Improved approximation algorithms for minimum-weight vertex separators

Abstract

We develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L1 (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes.We obtain an O( √ log n) approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be Θ( √ log n). We also prove various approximate max-flow/minvertex-cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n).These results have a number of applications. We exhibit an O( √ log n) pseudo-approximation for finding balanced vertex separators in general graphs. In fact, we achieve an approximation ratio of O( √ log opt) where opt is the size of an optimal separator, improving over the previous best bound of O(log opt). Likewise, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k √ log k), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor (which includes, e.g., bounded genus graphs), we give a constantfactor approximation for the treewidth; this can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.

Locality-sensitive and Re-use Promoting Personalized PageRank computations

“…Note that localities can be distance-constrained or size-constrained. Common definitions include h-hop neighborhoods (Boldi et al, 2011;Cohen et al, 2003;Wei, 2010;Xiao et al, 2009;Zhou et al, 2009), reachability neighborhoods (Cohen et al, 2003), cluster/partition neighborhoods (Feige et al, 2005;Karypis and Kumar, 1998;Newman, 2006), or hitting distance neighborhoods (Chen et al, 2008;Mei et al, 2008). One straight-forward way to identify the locality of a seed node n is to perform breadth-first search around n to locate the closest L nodes in linear time to the size of the locality.…”

Inapproximability of Treewidth and Related Problems

“…As mentioned earlier, determining the exact Treewidth of a graph and producing an associated optimal tree decomposition (see Definition 2.1) is known to be NP-hard (Arnborg et al, 1987), and a central open problem is to determine whether or not there exists a polynomial time constant factor approximation algorithm for Treewidth (see e.g., Bodlaender et al, 1995;Feige et al, 2005;Bodlaender, 2005). The current best polynomial time approximation algortihm for Treewidth (Feige et al, 2005), computes the Treewidth tw(G) within a factor O( log tw(G)).…”

“…The current best polynomial time approximation algortihm for Treewidth (Feige et al, 2005), computes the Treewidth tw(G) within a factor O( log tw(G)). On the other hand, the only hardness result to date for Treewidth shows that it is NP-hard to compute Treewidth within an additive error of n for some > 0 (Bodlaender et al, 1995).…”

“…And if there is a factor b approximation algorithm for Treewidth then there is an O(b log n) approximation algorithm for the related Pathwidth problem. The best currently known approximation factor for vertex separator is O( √ log n) (Feige, Hajiaghayi, & Lee, 2005) and thus the best algorithm for Treewidth finds a tree decomposition that is within an O( log n) factor of the optimal width, and an O(( √ log n)(log n)) factor approximation algorithm for Pathwidth.…”

Computing Pathwidth Faster Than 2 n

“…Some examples of efficient algorithms include polynomial time algorithms for Circle and Circular-arc graphs [15,18], and Permutation graphs [6], and fixed parameter tractable algorithms for both problems [3]. There also exist approximation algorithms for both problems, where Feige et al [8] give the most recent algorithm for treewidth.…”